30

Part of 2016 Online Math Open Problems

Problems(2)

2015-2016 OMO Spring #30

Source:

ABC

AB=3\sqrt{30}-\sqrt{10}

BC=12

CA=3\sqrt{30}+\sqrt{10}

M

be the midpoint of

AB

N

be the midpoint of

AC

l

as the line passing through the circumcenter

O

and orthocenter

H

ABC

E

F

be the feet of the perpendiculars from

B

C

l

, respectively. Let

l'

be the reflection of

l

BC

l'

intersects lines

AE

AF

P

Q

, respectively. Let lines

BP

CQ

K

X

Y

Z

are the reflections of

K

over the perpendicular bisectors of sides

BC

CA

AB

, respectively, and

R

S

are the midpoints of

XY

XZ

, respectively. If lines

MR

NS

T

, then the length of

OT

can be expressed in the form

\frac{p}{q}

for relatively prime positive integers

p

q

100p+q

.
Proposed by Vincent Huang and James Lin

Online Math Open

2016-2017 Fall OMO Problem 30

Source:

P_1(x),P_2(x),\ldots,P_n(x)

be monic, non-constant polynomials with integer coefficients and let

Q(x)

be a polynomial with integer coefficients such that

x^{2^{2016}}+x+1=P_1(x)P_2(x)\ldots P_n(x)+2Q(x).

Suppose that the maximum possible value of

2016n

can be written in the form

2^{b_1}+2^{b_2}+\cdots+2^{b_k}

for nonnegative integers

b_1<

b_2<

\cdots<

b_k

. Find the value of

b_1+b_2+\cdots+b_k

.
Proposed by Michael Ren

Online Math Open